Construction and analysis of degenerate PARAFAC models

By P. Paatero. 2000

125 citations recorded by the source corpus.

Subfields: Computational Mathematics, Computational Mechanics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/56b456e40c0d41c49a53df1bd60e4a57437fda53

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In Computational Mathematics

In Computational Mechanics

In Computational Theory and Mathematics

Citations in the corpus

Tensor Decomposition for Signal Processing and Machine Learning (2016, background): “This tensor has rank equal to 3, however it can be arbitrarily well approximated [23] by the following sequence of rank-two tensors (see also [14]):”
Tensor Rank and the Ill-Posedness of the Best Low-Rank Approximation Problem (2006, unknown): “[49, 51, 62, 67, 68]) but we prefer to avoid this term since it is inconsistent (and often at odds) with common usage in Mathematics.”
Tensors : A brief introduction (2014, unknown): “…form of a spatial prewhitening; (ii) impose orthogonality between decomposable tensors [45]; (iii) prevent divergence by bounding coefﬁcients λ r [61], [54]; (iv) if the tensor is nonnegative, use a nonnegative CP [54]; (v) impose a minimal angle between columns of factor matrices [55]; (vi)…”
Tensor decompositions, alternating least squares and other tales (2009, methodology): “For example, Paatero reports in Reference [16] precise examples where the trajectory from a well conditioned initial point to a well-deﬁned solution has to go around a region of higher rank, hence a ‘swamp’.”
Factorization strategies for third-order tensors (2011, background): “It may be that the best rank-k approximation does not exist [22, 35, 40].”
Local Convergence of the Alternating Least Squares Algorithm for Canonical Tensor Approximation (2012, background): “The great majority of the literature focuses on global properties of the iteration, like the existence of convergent subsequences and critical points, or the occurrence of swamps [3, 6, 7, 15, 17, 21, 22].”
Degeneracy in Candecomp/Parafac explained for p × p × 2 arrays of rank p + 1 or higher (2006, methodology): “ Paatero (2000) shows that these sequences of rank-2 arrays satisfy a particular Tucker3 model (3) with a 2 × 2 × 2 core array.”
A scalable optimization approach for fitting canonical tensor decompositions (2011, methodology): “In this paper we focus on the question of how to compute the CP factor matrices for a given value of R (not necessarily equal to the rank of the tensor).”
CANDECOMP/PARAFAC Decomposition of High-Order Tensors Through Tensor Reshaping (2012, background): “It is worth noting that, once again, CP solution may not exist [30], [31], [43], [8].”
Numerical construction of Green's functions in high dimensional elliptic problems with variable coefficients and analysis of renewable energy data via sparse and separable approximations (2012, unknown): “As an example, widely cited in the literature [42, 90, 87, 112], consider the tensor of nominal separation rank r = 3, A = a((1)) ⊗ b((1)) ⊗ c((2)) + a((1)) ⊗ b((2)) ⊗ c((1)) + a((2)) ⊗ b((1)) ⊗ c((1)), where the six vectors satisfy: 〈 a((1)),a((2)) 〉 , 〈 b((1)),b((2)) 〉 , 〈 c((1)), c((2)) 〉 6= 0.”